Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $r = \dfrac{5k^2 - 50k}{k - 2} \times \dfrac{k - 9}{k^2 - 19k + 90} $
Answer: First factor the quadratic. $r = \dfrac{5k^2 - 50k}{k - 2} \times \dfrac{k - 9}{(k - 10)(k - 9)} $ Then factor out any other terms. $r = \dfrac{5k(k - 10)}{k - 2} \times \dfrac{k - 9}{(k - 10)(k - 9)} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac{ 5k(k - 10) \times (k - 9) } { (k - 2) \times (k - 10)(k - 9) } $ $r = \dfrac{ 5k(k - 10)(k - 9)}{ (k - 2)(k - 10)(k - 9)} $ Notice that $(k - 9)$ and $(k - 10)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac{ 5k\cancel{(k - 10)}(k - 9)}{ (k - 2)\cancel{(k - 10)}(k - 9)} $ We are dividing by $k - 10$ , so $k - 10 \neq 0$ Therefore, $k \neq 10$ $r = \dfrac{ 5k\cancel{(k - 10)}\cancel{(k - 9)}}{ (k - 2)\cancel{(k - 10)}\cancel{(k - 9)}} $ We are dividing by $k - 9$ , so $k - 9 \neq 0$ Therefore, $k \neq 9$ $r = \dfrac{5k}{k - 2} ; \space k \neq 10 ; \space k \neq 9 $